Size Distribution Of Cities

Journal of Urban Economics 57 (2005) 86–100 www.elsevier.com/locate/jue

Zipf zipped
Volker Nitsch
Department of Economics, Free University Berlin, Boltzmannstrasse 20, 14195 Berlin, Germany Received 8 March 2004; revised 30 August 2004 Available online 24 November 2004

Abstract In this paper, I provide a quantitative review of the empirical literature on Zipf’s law for cities; themeta-analysis combines 515 estimates from 29 studies. I find that the combined estimate of the Zipf coefficient is significantly larger than 1.0. This finding implies that cities are on average more evenly distributed than suggested by (a strict interpretation of) Zipf’s law. I also identify several features that account for differences across the individual point estimates.  2004 Elsevier Inc. All rightsreserved.
JEL classification: R12; R15 Keywords: Zipf’s law; Size distribution of cities; Rank size rule; Meta-analysis

1. Introduction In 1913, the German geographer Felix Auerbach described an interesting empirical regularity: analyzing the size distribution of cities, he found that the product of the population size of a city and its rank in the distribution appears to be roughly constant fora given territory. Thus, the second-largest city has on average about one-half the population of the largest city, the number 3 city one-third that population, and the number n city 1/n that population. Since there is no obvious reason why the hierarchy of cities should follow such a pattern, this rank–size rule, also known as Zipf’s law for cities after George Zipf [22], has attractedconsiderable interest.
E-mail address: vnitsch@wiwiss.fu-berlin.de. 0094-1190/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jue.2004.09.002

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The standard approach to explore city size distributions is simple and intuitive. Based on Auerbach’s [1] proposition that Pi Ri = A, with Pi the population size of cityi, Ri its rank, and A a positive constant, the typical regression takes the form: ln Ri = ln A − α ln Pi . Tests of Zipf’s law then include whether this equation describes the city size distribution reasonably well,1 the estimated coefficient α is close to one,2 and A corresponds to the size of the largest city. While numerous studies have applied this regression for various territories and timeperiods, the interpretation of the results is ambiguous. Some authors emphasize the strikingly good empirical fit of the log-linear rank–size relationship. Krugman [14, p. 40], for instance, argues that “[w]e are unused to seeing regularities this exact in economics.” Gabaix [6, p. 739] notes that “Zipf’s law for cities is one of the most conspicuous empirical facts in economics, or in the socialsciences generally.” Others take a much more skeptical view. Sheppard [19, p. 131], for instance, questions whether the rank–size rule is the best possible representation of city size distributions; other distributional functions may conform more closely to the data. Henderson [10, p. 391] argues that “[i]n general and on average, the rank–size rule simply does not hold.” In this paper, I follow adifferent approach. Instead of providing another test of Zipf’s law, I perform a meta-analysis of estimated Zipf coefficients. That is, I use a set of statistical techniques to combine and evaluate the results from other studies.3 This approach offers several advantages. First, it provides a summary estimate of the parameter of interest, Zipf coefficient α, for a wide range of different data sets andmethodologies. Derived from a systematic aggregation of existing research evidence, this estimate appears to be much more credible and accurate than any finding from an individual study. Second, metaanalysis allows to explore the sensitivity of the estimate to characteristics of the underlying study. While deviations from Zipf’s law are often attributed to variations in the estimation procedure,...